Classical Fourier Analysis

Graduate Texts in Mathematics

Classical

Fourier

Analysis

Loukas Grafakos

Third Edition

Graduate Texts in Mathematics 249

Graduate Texts in Mathematics

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San Francisco State University, San Francisco, CA, USA

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Loukas Grafakos

Classical Fourier Analysis

Third Edition

123

Loukas Grafakos

Department of Mathematics

University of Missouri

Columbia, MO, USA

ISSN 0072-5285 ISSN 2197-5612 (electronic)

ISBN 978-1-4939-1193-6 ISBN 978-1-4939-1194-3 (eBook)

DOI 10.1007/978-1-4939-1194-3

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To Suzanne

Preface

The great response to the publication of my book Classical and Modern Fourier

Analysis in 2004 has been especially gratifying to me. I was delighted when Springer

offered to publish the second edition in 2008 in two volumes: Classical Fourier

Analysis, 2nd Edition, and Modern Fourier Analysis, 2nd Edition. I am now elated

to have the opportunity to write the present third edition of these books, which

Springer has also kindly offered to publish. The third edition was born from my

desire to improve the exposition in several places, fix a few inaccuracies, and add

some new material. I have been very fortunate to receive several hundred e-mail

messages that helped me improve the proofs and locate mistakes and misprints in

the previous editions.

In this edition, I maintain the same style as in the previous ones. The proofs con￾tain details that unavoidably make the reading more cumbersome. Although it will

behoove many readers to skim through the more technical aspects of the presenta￾tion and concentrate on the flow of ideas, the fact that details are present will be

comforting to some. (This last sentence is based on my experience as a graduate

student.) Readers familiar with the second edition will notice that the chapter on

weights has been moved from the second volume to the first.

This first volume Classical Fourier Analysis is intended to serve as a text for

a one-semester course with prerequisites of measure theory, Lebesgue integration,

and complex variables. I am aware that this book contains significantly more ma￾terial than can be taught in a semester course; however, I hope that this additional

information will be useful to researchers. Based on my experience, the following list

of sections (or parts of them) could be taught in a semester without affecting the

logical coherence of the book: Sections 1.1, 1.2, 1.3, 2.1, 2.2., 2.3, 3.1, 3.2, 3.3, 4.4,

4.5, 5.1, 5.2, 5.3, 5.5, 5.6, 6.1, 6.2.

A long list of people have assisted me in the preparation of this book, but I remain

solely responsible for any misprints, mistakes, and omissions contained therein.

Please contact me directly ([email protected]) if you have corrections or com￾ments. Any corrections to this edition will be posted to the website

http://math.missouri.edu/˜loukas/FourierAnalysis.html

vii

viii Preface

which I plan to update regularly. I have prepared solutions to all of the exercises for

the present edition which will be available to instructors who teach a course out of

this book.

Athens, Greece, Loukas Grafakos

March 2014

Acknowledgments

I am extremely fortunate that several people have pointed out errors, misprints, and

omissions in the previous editions of the books in this series. All these individuals

have provided me with invaluable help that resulted in the improved exposition of

the text. For these reasons, I would like to express my deep appreciation and sincere

gratitude to the all of the following people.

ix

First edition acknowledgements: Georgios Alexopoulos, Nakhle Asmar, Bruno ´

Calado, Carmen Chicone, David Cramer, Geoffrey Diestel, Jakub Duda, Brenda

Frazier, Derrick Hart, Mark Hoffmann, Steven Hofmann, Helge Holden, Brian

Hollenbeck, Petr Honz´ık, Alexander Iosevich, Tunde Jakab, Svante Janson, Ana

Jimenez del Toro, Gregory Jones, Nigel Kalton, Emmanouil Katsoprinakis, Dennis ´

Kletzing, Steven Krantz, Douglas Kurtz, George Lobell, Xiaochun Li, Jose Mar ´ ´ıa

Martell, Antonios Melas, Keith Mersman, Stephen Montgomery-Smith, Andrea

Nahmod, Nguyen Cong Phuc, Krzysztof Oleszkiewicz, Cristina Pereyra, Carlos

Perez, Daniel Redmond, Jorge Rivera-Noriega, Dmitriy Ryabogin, Christopher ´

Sansing, Lynn Savino Wendel, Shih-Chi Shen, Roman Shvidkoy, Elias M. Stein,

Atanas Stefanov, Terence Tao, Erin Terwilleger, Christoph Thiele, Rodolfo Torres,

Deanie Tourville, Nikolaos Tzirakis, Don Vaught, Igor Verbitsky, Brett Wick, James

Wright, and Linqiao Zhao.

Second edition acknowledgements: Marco Annoni, Pascal Auscher, Andrew

Bailey, Dmitriy Bilyk, Marcin Bownik, Juan Cavero de Carondelet Fiscowich,

Leonardo Colzani, Simon Cowell, Mita Das, Geoffrey Diestel, Yong Ding, Jacek

Dziubanski, Frank Ganz, Frank McGuckin, Wei He, Petr Honz´ık, Heidi Hulsizer,

Philippe Jaming, Svante Janson, Ana Jimenez del Toro, John Kahl, Cornelia Kaiser, ´

Nigel Kalton, Kim Jin Myong, Doowon Koh, Elena Koutcherik, David Kramer,

Enrico Laeng, Sungyun Lee, Qifan Li, Chin-Cheng Lin, Liguang Liu, Stig-Olof

Londen, Diego Maldonado, Jose Mar ´ ´ıa Martell, Mieczysław Mastyło, Parasar

Mohanty, Carlo Morpurgo, Andrew Morris, Mihail Mourgoglou, Virginia Naibo,

Tadahiro Oh, Marco Peloso, Maria Cristina Pereyra, Carlos Perez, Humberto Rafeiro, ´

Maria Carmen Reguera Rodr´ıguez, Alexander Samborskiy, Andreas Seeger, Steven

Senger, Sumi Seo, Christopher Shane, Shu Shen, Yoshihiro Sawano, Mark Spencer,

Vladimir Stepanov, Erin Terwilleger, Rodolfo H. Torres, Suzanne Tourville,

x Acknowledgments

Among all these people, I would like to give special thanks to an individual who

has studied extensively the two books in the series and has helped me more than

anyone else in the preparation of the third edition: Danqing He. I am indebted to him

for all the valuable corrections, suggestions, and constructive help he has provided

me with in this work. Without him, these books would have been a lot poorer.

Finally, I would also like to thank the University of Missouri for granting me

a research leave during the academic year 2013-2014. This time off enabled me to

finish the third edition of this book on time. I spent my leave in Greece.

Ignacio Uriarte-Tuero, Kunyang Wang, Huoxiong Wu, Kozˆ o Yabuta, Takashi ˆ

Yamamoto, and Dachun Yang.

Third edition acknowledgments: Marco Annoni, Mark Ashbaugh, Daniel

Azagra, Andrew Bailey, Arpad B ´ enyi, Dmitriy Bilyk, Nicholas Boros, Almut ´

Burchard, Mar´ıa Carro, Jameson Cahill, Juan Cavero de Carondelet Fiscowich,

Xuemei Chen, Andrea Fraser, Shai Dekel, Fausto Di Biase, Zeev Ditzian, Jianfeng

Dong, Oliver Dragicevi ˇ c, Sivaji Ganesh, Friedrich Gesztesy, Zhenyu Guo, Piotr ´

Hajłasz, Danqing He, Andreas Heinecke, Steven Hofmann, Takahisa Inui, Junxiong

Jia, Kasinathan Kamalavasanthi, Hans Koelsch, Richard Laugesen, Kaitlin Leach,

Andrei Lerner, Yiyu Liang, Calvin Lin, Liguang Liu, Elizabeth Loew, Chao Lu,

Richard Lynch, Diego Maldonado, Lech Maligranda, Richard Marcum, Mieczysław

Mastyło, Mariusz Mirek, Carlo Morpurgo, Virginia Naibo, Hanh Van Nguyen,

Seungly Oh, Tadahiro Oh, Yusuke Oi, Lucas da Silva Oliveira, Kevin O’Neil, Hesam

Oveys, Manos Papadakis, Marco Peloso, Carlos Perez, Jesse Peterson, Dmitry ´

Prokhorov, Amina Ravi, Maria Carmen Reguera Rodr´ıguez, Yoshihiro Sawano,

Mirye Shin, Javier Soria, Patrick Spencer, Marc Strauss, Krystal Taylor, Naohito

Tomita, Suzanne Tourville, Rodolfo H. Torres, Fujioka Tsubasa, Ignacio Uriarte￾Tuero, Brian Tuomanen, Shibi Vasudevan, Michael Wilson, Dachun Yang, Kai Yang,

Yandan Zhang, Fayou Zhao, and Lifeng Zhao.

Contents

1 Lp Spaces and Interpolation 1

1.1 Lp and Weak Lp ............................................ 1

1.1.1 The Distribution Function ............................. 3

1.1.2 Convergence in Measure .............................. 6

1.1.3 A First Glimpse at Interpolation ........................ 9

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.2 Convolution and Approximate Identities . . . . . . . . . . . . . . . . . . . . . . . 17

1.2.1 Examples of Topological Groups . . . . . . . . . . . . . . . . . . . . . . . 18

1.2.2 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.2.3 Basic Convolution Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.2.4 Approximate Identities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

1.3 Interpolation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

1.3.1 Real Method: The Marcinkiewicz Interpolation Theorem . . . 33

1.3.2 Complex Method: The Riesz–Thorin Interpolation

Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

1.3.3 Interpolation of Analytic Families of Operators . . . . . . . . . . . 40

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

1.4 Lorentz Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

1.4.1 Decreasing Rearrangements . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

1.4.2 Lorentz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

1.4.3 Duals of Lorentz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

1.4.4 The Off-Diagonal Marcinkiewicz Interpolation Theorem . . . 60

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

2 Maximal Functions, Fourier Transform, and Distributions 85

2.1 Maximal Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

2.1.1 The Hardy–Littlewood Maximal Operator. . . . . . . . . . . . . . . . 86

2.1.2 Control of Other Maximal Operators . . . . . . . . . . . . . . . . . . . . 90

xi

xii Contents

2.1.3 Applications to Differentiation Theory . . . . . . . . . . . . . . . . . . 93

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

2.2 The Schwartz Class and the Fourier Transform . . . . . . . . . . . . . . . . . . 104

2.2.1 The Class of Schwartz Functions . . . . . . . . . . . . . . . . . . . . . . . 105

2.2.2 The Fourier Transform of a Schwartz Function . . . . . . . . . . . 108

2.2.3 The Inverse Fourier Transform and Fourier Inversion . . . . . . 111

2.2.4 The Fourier Transform on L1 +L2 . . . . . . . . . . . . . . . . . . . . . . 113

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

2.3 The Class of Tempered Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 119

2.3.1 Spaces of Test Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

2.3.2 Spaces of Functionals on Test Functions . . . . . . . . . . . . . . . . . 120

2.3.3 The Space of Tempered Distributions . . . . . . . . . . . . . . . . . . . 123

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

2.4 More About Distributions and the Fourier Transform . . . . . . . . . . . . . 133

2.4.1 Distributions Supported at a Point . . . . . . . . . . . . . . . . . . . . . . 134

2.4.2 The Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

2.4.3 Homogeneous Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

2.5 Convolution Operators on Lp Spaces and Multipliers . . . . . . . . . . . . . 146

2.5.1 Operators That Commute with Translations . . . . . . . . . . . . . . 146

2.5.2 The Transpose and the Adjoint of a Linear Operator . . . . . . . 150

2.5.3 The Spaces M p,q(Rn) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

2.5.4 Characterizations of M1,1(Rn) and M2,2(Rn). . . . . . . . . . . . 153

2.5.5 The Space of Fourier Multipliers Mp(Rn) . . . . . . . . . . . . . . . 155

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

2.6 Oscillatory Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

2.6.1 Phases with No Critical Points . . . . . . . . . . . . . . . . . . . . . . . . . 161

2.6.2 Sublevel Set Estimates and the Van der Corput Lemma . . . . . 164

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

3 Fourier Series 173

3.1 Fourier Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

3.1.1 The n-Torus Tn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

3.1.2 Fourier Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

3.1.3 The Dirichlet and Fejer Kernels ´ . . . . . . . . . . . . . . . . . . . . . . . . 178

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

3.2 Reproduction of Functions from Their Fourier Coefficients. . . . . . . . 183

3.2.1 Partial sums and Fourier inversion . . . . . . . . . . . . . . . . . . . . . . 183

3.2.2 Fourier series of square summable functions . . . . . . . . . . . . . 185

3.2.3 The Poisson Summation Formula . . . . . . . . . . . . . . . . . . . . . . . 187

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

3.3 Decay of Fourier Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

3.3.1 Decay of Fourier Coefficients of Arbitrary Integrable

Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

3.3.2 Decay of Fourier Coefficients of Smooth Functions. . . . . . . . 195

Contents xiii

3.3.3 Functions with Absolutely Summable Fourier

Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

3.4 Pointwise Convergence of Fourier Series . . . . . . . . . . . . . . . . . . . . . . . 204

3.4.1 Pointwise Convergence of the Fejer Means ´ . . . . . . . . . . . . . . . 204

3.4.2 Almost Everywhere Convergence of the Fejer Means ´ . . . . . . 207

3.4.3 Pointwise Divergence of the Dirichlet Means . . . . . . . . . . . . . 210

3.4.4 Pointwise Convergence of the Dirichlet Means. . . . . . . . . . . . 212

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

3.5 A Tauberian theorem and Functions of Bounded Variation . . . . . . . . 216

3.5.1 A Tauberian theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

3.5.2 The sine integral function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

3.5.3 Further properties of functions of bounded variation . . . . . . . 219

3.5.4 Gibbs phenomenon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

3.6 Lacunary Series and Sidon Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

3.6.1 Definition and Basic Properties of Lacunary Series . . . . . . . . 227

3.6.2 Equivalence of Lp Norms of Lacunary Series . . . . . . . . . . . . . 229

3.6.3 Sidon sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

4 Topics on Fourier Series 241

4.1 Convergence in Norm, Conjugate Function,

and Bochner–Riesz Means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

4.1.1 Equivalent Formulations of Convergence in Norm . . . . . . . . . 242

4.1.2 The Lp Boundedness of the Conjugate Function . . . . . . . . . . . 246

4.1.3 Bochner–Riesz Summability . . . . . . . . . . . . . . . . . . . . . . . . . . . 250

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

4.2 A. E. Divergence of Fourier Series and Bochner–Riesz means . . . . . 255

4.2.1 Divergence of Fourier Series of Integrable Functions . . . . . . 255

4.2.2 Divergence of Bochner–Riesz Means of Integrable

Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270

4.3 Multipliers, Transference, and Almost Everywhere Convergence . . . 271

4.3.1 Multipliers on the Torus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

4.3.2 Transference of Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

4.3.3 Applications of Transference . . . . . . . . . . . . . . . . . . . . . . . . . . . 280

4.3.4 Transference of Maximal Multipliers . . . . . . . . . . . . . . . . . . . . 281

4.3.5 Applications to Almost Everywhere Convergence . . . . . . . . . 285

4.3.6 Almost Everywhere Convergence of Square Dirichlet

Means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

xiv Contents

4.4 Applications to Geometry and Partial Differential Equations. . . . . . . 292

4.4.1 The Isoperimetric Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

4.4.2 The Heat Equation with Periodic Boundary Condition . . . . . 294

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

4.5 Applications to Number theory and Ergodic theory . . . . . . . . . . . . . . 299

4.5.1 Evaluation of the Riemann Zeta Function at even Natural

numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

4.5.2 Equidistributed sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302

4.5.3 The Number of Lattice Points inside a Ball . . . . . . . . . . . . . . . 305

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308

5 Singular Integrals of Convolution Type 313

5.1 The Hilbert Transform and the Riesz Transforms . . . . . . . . . . . . . . . . 313

5.1.1 Definition and Basic Properties of the Hilbert Transform . . . 314

5.1.2 Connections with Analytic Functions. . . . . . . . . . . . . . . . . . . . 317

5.1.3 Lp Boundedness of the Hilbert Transform . . . . . . . . . . . . . . . . 319

5.1.4 The Riesz Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

5.2 Homogeneous Singular Integrals and the Method of Rotations . . . . . 333

5.2.1 Homogeneous Singular and Maximal Singular Integrals . . . . 333

5.2.2 L2 Boundedness of Homogeneous Singular Integrals . . . . . . 336

5.2.3 The Method of Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339

5.2.4 Singular Integrals with Even Kernels. . . . . . . . . . . . . . . . . . . . 341

5.2.5 Maximal Singular Integrals with Even Kernels . . . . . . . . . . . 347

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353

5.3 The Calderon–Zygmund Decomposition and Singular Integrals ´ . . . . 355

5.3.1 The Calderon–Zygmund Decomposition ´ . . . . . . . . . . . . . . . . . 355

5.3.2 General Singular Integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358

5.3.3 Lr Boundedness Implies Weak Type (1,1) Boundedness. . . . 359

5.3.4 Discussion on Maximal Singular Integrals . . . . . . . . . . . . . . . 362

5.3.5 Boundedness for Maximal Singular Integrals Implies

Weak Type (1,1) Boundedness. . . . . . . . . . . . . . . . . . . . . . . . . 366

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371

5.4 Sufficient Conditions for Lp Boundedness . . . . . . . . . . . . . . . . . . . . . . 374

5.4.1 Sufficient Conditions for Lp Boundedness of Singular

Integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375

5.4.2 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378

5.4.3 Necessity of the Cancellation Condition . . . . . . . . . . . . . . . . . 379

5.4.4 Sufficient Conditions for Lp Boundedness of Maximal

Singular Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384

5.5 Vector-Valued Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385

5.5.1 2-Valued Extensions of Linear Operators . . . . . . . . . . . . . . . . 386

5.5.2 Applications and r

-Valued Extensions of Linear

Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390